The elastic energy stored in your tendons can contribute up to 35% of your energy needs when running. Sports scientists find that (on average) the knee extensor tendons in sprinters stretch 41 mm while those of nonathletes stretch only 33 mm. The spring constant of the tendon is the same for both groups, 33 N/mm. What is the difference in maximum stored energy between the sprinters and the nonathletes?

Video transcript

Hey, everyone in this problem, we're told that elastic cords are used to store and release energy. The stretching of a cord is determined by the design of the equipment. One equipment design allows a maximum stretching of 5.2 cm while another allows a maximum stretching of 8.2 cm. Both types of equipment are fitted with the same type of cord having a force constant of 25 newtons per centimeter. We're asked to calculate the difference In the greatest energy stored by the cords in the two equipment designs. We're given four answer choices. option a 503 jewels, Option B 5.03 jewels, option C 37.5 jewels and option D 0.375 jewels. Now we have an elastic chord. Ok. So let's recall it. The energy in an elastic chord is given by one half, multiplied by K, multiplied by X squared where K is the force constant and X is the distance that we've stretched or compressed. So we want to find the difference in the greatest energy for these two equipment designs. So let's start with the first equipment design. We're gonna call this E one, the energy of the first equipment design, This is gonna be equal to 1/2 Multiplied by K one multiplied by X one squared. Now, we're asked to find the greatest energy, OK. So we actually wanna find the maximum value of E one E one max. OK. This is gonna be equal to one half multiplied by K1 which is a constant multiplied by X one max squared. OK. The maximum stretch now we're given the force constant K, we're given the maximum stretch. So we can go ahead and substitute in these values to figure out the maximum energy stored in this first core design. So this is gonna be 1/ multiplied By 2500 newtons per meter. OK. multiplied by 0.05, two m all squared. Now, how did we get these numbers? OK. In the problem, we were given 25 newtons per centimeter for our force constant. We want to write this in our standard unit newtons per meter. OK. So in order to convert, we multiply, we know that there are 100cm per meter, OK? The unit of centimeter will divide out. We're left with 25 multiplied by 100 Which gives us 2500 newtons per meter. And similarly, for the maximum stretch 5. cm We have 5.2 cm, we're gonna multiply And we know in every one m there are 100 cm. So we divide by 100 The unit of centimeter cancels and we get 0. m. OK. So that's how we got those values just by converting our units. If we work this out, this gives us a maximum energy of 3.3, 8 jewels. Ok. So design one maximum energy of 3. jos what about for design two? For design two, we're gonna do the same thing. The energy of design two is one half multiplied by K two, multiplied by X two squared. We wanna find the maximum. So we have E two max is gonna be one half multiplied by K two. OK. That's a constant. We can't change that value Multiplied by X two Max that maximum stretch. Now, in this case, our maximum stretch that we're given is 8.2 cm. We can do the exact same thing As we did with 5.2 cm. We divide by 100, The unit of centimeter cancels and we get 0.08, two m. Now, both types of equipment have this same force constant. So the value of K two is the same as K 1 25 newtons per meter. So our E two max is 1/2 Multiplied by 2500 newtons per meter Multiplied by 0.082 m all squared. And when we work this out, we get a maximum energy In co two or design two of 8.405 jewels. So we found the maximum amount of energy that can be stored in each of these designs. And now we're asked to find the difference, OK? And when we're talking about the difference, we're just talking about the magnitude of the difference. So we're gonna take the absolute value, the difference in energy is going to be equal to the absolute value of E two max -11 Max. OK? In this case, E two Max is larger, so we can drop that absolute value sign. So we have 8.405 jewels -3.38 Jews. And this gives us an energy difference of 5.025 Jews. And that's the answer we were looking for if we compare this to our answer choices, OK? We can see that we have three significant digits. We found that the difference in the greatest energy stored by the chords in the two equipment designs is 5.03 jewels which corresponds with answer choice. B thanks everyone for watching. I hope this video helped see you in the next one.